 UPSKILL MATH PLUS

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### Theory:

1. If a prime number $$p$$ divides $$ab$$, then $$p$$ divides either $$a$$ or $$b$$. That is, $$p$$ divides at least one of them.
Example:
Let us take a prime number $$3$$ divides $$5 \times 6$$.

$\frac{5×6}{3}$

Here, $$3$$ cannot divide $$5$$, but it divides $$6$$.

That is, a prime number $$p$$ divides at least one of them.

2. If a composite number $$n$$ divides $$ab$$, then $$n$$ neither divide $$a$$ nor $$b$$.
Example:
Let us take a  composite number $$4$$ divides $$2 \times 6$$.

$\frac{2×6}{4}$

Here, $$4$$ neither divide $$2$$ nor divide $$6$$. But, it divides the product of $$2 \times 6 = 12$$.

Thus, if a composite number $$n$$ divides $$ab$$, then $$n$$ neither divide $$a$$ nor $$b$$.
Fun Fact
The six-digit number of the form $$xyxyxy$$ (where $$1 \le x \le 9, 1 \le y \le 9$$) always divisible by the number $$10101$$.

Explanation:

$$xyxyxy = (xy \times 10000) + (xy \times 100) + xy$$

$$xyxyxy = xy (10000 + 100 + 1)$$

$$xyxyxy = xy (10101)$$

$\frac{\mathit{xyxyxy}}{10101}=\mathit{xy}$

Thus, any six-digit number of the form $$xyxyxy$$ (where $$1 \le x \le 0, 1 \le y \le 9$$) always divisible by the number $$10101$$.